Making inequalities interesting

Question

Well, I'm learning olympiad inequalities, and most of the books go like this:

Memorize holder-cauchy-jensen-muirhead, then just dumbass (i.e bash and homogenize) and use the above theorems.

Needless to say, I feel extremely disintersted in inequalities for this approach.

What are some good books which makes inequality interesting ? (Not involving extremely elementary things like AM-GM, Cauchy, but more advanced like Jensen-Muirhead-Karamata-Schur also) ?

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To give clarification what I consider as intersting as of now, the only inequality problem that I found interseting is that the following statement for positive reals

$\displaystyle \sum_{i=0}^{n}\frac{a_i}{a_{i-1}+a_{[i+1 \mod n}]} \leq \frac{1}{2}$

only holds for finitely many $n$. Also one thing that I find interesting is that the bounds which can't be analytically found, but exists.

Answer 1

Perhaps that you've already been exposed to it, but the great classical reference is Inequalities, by Hardy, Littlewood, and Pólya.

Answer 2

Because for $a_i=1$ we have $$\frac{n+1}{2}\leq\frac{1}{2},$$ which is wrong sometimes.

I think, a best way to learn inequalities today it's to read books and papers of Vasile Cirtoaje.